284 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390
and on integration over angular space,
(23)
(24)
Introducing the value of A2 from (19), we have finally
277-r2 Mod2 \/r. sin & dd =
" IcR
If we suppose that kR is large, but still so that R is small compared with r, (24) reduces to S-Tr/c"2 or 2V/7T. The energy dispersed is then the double of that which would be dispersed by each resonator acting alone; otherwise the mutual reaction complicates the expression.
The greatest interference naturally occurs when kR is small. (24) then becomes 2/e2.R2. 2V/TT, or IQ^rR-, in agreement with Theory of Sound, § 321. The whole energy dispersed is then much less than if there were only one resonator.
It is of interest to trace the influence of distance more closely. If we put kR — 2-Trm, so that R — m\ we may write (24)
O — / 9 "\ 2 / mA 7P {ty^l\
where S is the area of primary wave-front which carries the same energy as is dispersed by the two resonators and
„ 27TTO + sin (2-Trm)
2-Trw + (2-Trm)-1 -f 2 sin(2-7rm) If 2m is an integer, the sine vanishes and
1
.(26)
.(27)
not differing much from unity even when 2m = 1; and whenever 2m is great, F approaches unity.
The following table gives the values of F for values of 2m not greater than 2:
2m F 2m F 2m F
0-05 0-0459 0-70 0-7042 1-40 1-266
o-io 0-1514 0-80 0-7588 1-50 1-269
0-20 0-3582 0-90 0-8256 1-60 1-226
0-30 0-4836 1-00 0-9080 1-70 1-159
0-40 0-5583 1-10 1-006 1-80 1-088
0-50 0-6110 1-20 1-113 1-90 1-026
0-60 0-6569 1-30 1-208 2-00 0-975